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Unformatted text preview: + 2 [ ej(9t-tl + e -J(9t-t l] = 16 + 12cos (3t - ~) + Bcos (6t -~) + 4 cos (9t - ~) Clearly b oth sets of spectra represent t he same periodic signal. • S ignal R epresentation b y O rthogonal S ets 3 212 3.5 E xponential F ourier S eries 213 Bandwidth of a Signal T he d ifference b etween t he h ighest a nd t he lowest frequencies o f t he s pectral
c omponents o f a s ignal is t he b andwidth o f t he s ignal. T he b andwidth o f t he
s ignal w hose e xponential s pectra a re s hown i n F ig. 3.15b is 9 ( in r adians). T he
h ighest a nd l owest frequencies a re 9 a nd 0 r espectively. N ote t hat t he c omponent o f
f requency 12 h as z ero a mplitude a nd is n onexistent. M oreover, t he l owest frequency
is 0, n ot - 9. R ecall t hat t he f requencies (in t he c onventional sense) o f t he s pectral
c omponents a t w = - 3, -6, a nd - 9 i n r eality a re 3, 6, a nd 9 .t T he b andwidth c an
b e m ore r eadily s een f rom t he t rigonometric s pectra i n Fig. 3.15a. ••• • ••
o -27;, ••• • •• E xample 3 .8
F ind t he exponential Fourier series and sketch t he corresponding spectra for t he
impulse train OTo ( t) depicted in Fig. 3.16a. From this result sketch t he trigonometric
spectrum and write the trigonometric Fourier series for OTo (t).
T he e xponential Fourier series is given by (a) 27;, (b) • 211' W o=- To n =-OO where
Dn = ~
To Choosing t he i nterval of integration
OTo(t) = o (t), j Figure 3.16e shows the trigonometric Fourier spectrum. From this spectrum we can express
OTo (t) as T O/2 o (t)e-jnwot dt OTo(t) = To f ejnwot n =-OO 211' w o=- To (3.80) Equation (3.79) shows t hat t he exponential spectrum is uniform ( Dn = l iTo) for all the
frequencies, as shown in Fig. 3.16b. T he s pectrum, being real, requires only t he a mplitude
plot. All phases a re zero.
To sketch t he trigonometric spectrum, we use Eq. (3.77) to obtain
Co = Do = 1
To C n = 21Dnl = - On (e) F ig. 3 .16 Impulse t rain a nd its Fourier spectra. a nd recognizing t hat over this interval - To/2 ~ ••• o OTo(t)e-jnwot dt S ubstitution of t his value in Eq. (3.78) yields t he desired exponential Fourier series To To To In this integral t he impulse is located a t t = O. From the sampling property (1.24a),
t he integral on t he right-hand side is t he value of e-jnwot a t t = 0 (where the impulse is
Dn=~ OTO (t) = ..!. (3.78) 1 (=?' If) Dn = ~
To e" 2 [1 + 2(cos wot + cos 2wot + cos 3wot + ... )]
'0 = 1 ,2,3,··· =0 tSome authors do define bandwidth as the difference between the highest and the lowest (negative)
frequency in the exponential spectrum. The bandwidth according to this definition is twice that
defined here. In reality, this definition defines not the signal bandwidth but the s pectral w idth
(width of the exponential spectrum of the signal). n Effect of Symmetry in Exponential Fourier Series (3.81) • W hen j (t) h as a n e ven s ymmetry, bn = 0, a nd f rom E q. ( 3.74), D n = a nl2,
w hich is r eal ( positive o r n egative). H ence, L Dn c an o nly b e 0 o r ±11'. M...
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